Question 1

We will construct a triangle, find its circumcenter, and then draw its circumcircle. We will also determine if the circumcenter is inside or outside the triangle.

Step 1 — Construct Triangle ABC

We start by drawing the base. Let's draw a line segment AB. Its length is 5 cm. At point A, we construct an angle. This angle is $\angle\text{CAB} = \textbf{70}^\circ$. At point B, we construct another angle. This angle is $\angle\text{CBA} = \textbf{60}^\circ$. The rays from A and B meet. They meet at point C. Now, $\Delta\text{ABC}$ is formed.

Step 2 — Find Circumcentre O

Now, let's find the circumcenter. We draw the perpendicular bisector of AB. We draw the perpendicular bisector of BC. We also draw the perpendicular bisector of AC. These three bisectors meet at one point. Let's call this point O. Point O is the circumcenter of $\Delta\text{ABC}$.

Step 3 — Draw Circumcircle

We use point O as the center. The radius will be OA. We can also use OB or OC. Let's draw a circle with center O. The circle passes through A, B, and C. This is the circumcircle of $\Delta\text{ABC}$.

Step 4 — Determine Circumcentre Position

Let's find the third angle of the triangle. The sum of angles in a triangle is $\textbf{180}^\circ$. So, $\angle\text{C} = 180^\circ - (\angle\text{A} + \angle\text{B})$.

$\angle\text{C} = 180^\circ - (70^\circ + 60^\circ)$

$= 180^\circ - 130^\circ$

$\boxed{50^\circ}$

All angles are less than $\textbf{90}^\circ$. $\angle\text{A} = \textbf{70}^\circ$. $\angle\text{B} = \textbf{60}^\circ$. $\angle\text{C} = \textbf{50}^\circ$. This means $\Delta\text{ABC}$ is an acute-angled triangle. For an acute-angled triangle, the circumcenter is always inside.

Answer

The centre is inside the triangle.